Optimal. Leaf size=133 \[ -\frac {3 x^4 \left (1-x^2\right )^{2/3}}{10 \left (3+x^2\right )}+\frac {9 \left (1-x^2\right )^{2/3} \left (69+14 x^2\right )}{40 \left (3+x^2\right )}+\frac {99 \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{8\ 2^{2/3}}-\frac {99 \log \left (3+x^2\right )}{16\ 2^{2/3}}+\frac {297 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {457, 102, 151,
57, 631, 210, 31} \begin {gather*} \frac {99 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{8\ 2^{2/3}}+\frac {9 \left (1-x^2\right )^{2/3} \left (14 x^2+69\right )}{40 \left (x^2+3\right )}-\frac {99 \log \left (x^2+3\right )}{16\ 2^{2/3}}+\frac {297 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}-\frac {3 \left (1-x^2\right )^{2/3} x^4}{10 \left (x^2+3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 102
Rule 151
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{1-x} (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac {3 x^4 \left (1-x^2\right )^{2/3}}{10 \left (3+x^2\right )}-\frac {3}{10} \text {Subst}\left (\int \frac {x (-6+7 x)}{\sqrt [3]{1-x} (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac {3 x^4 \left (1-x^2\right )^{2/3}}{10 \left (3+x^2\right )}+\frac {9 \left (1-x^2\right )^{2/3} \left (69+14 x^2\right )}{40 \left (3+x^2\right )}+\frac {99}{8} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {3 x^4 \left (1-x^2\right )^{2/3}}{10 \left (3+x^2\right )}+\frac {9 \left (1-x^2\right )^{2/3} \left (69+14 x^2\right )}{40 \left (3+x^2\right )}-\frac {99 \log \left (3+x^2\right )}{16\ 2^{2/3}}+\frac {297}{16} \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac {297 \text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ &=-\frac {3 x^4 \left (1-x^2\right )^{2/3}}{10 \left (3+x^2\right )}+\frac {9 \left (1-x^2\right )^{2/3} \left (69+14 x^2\right )}{40 \left (3+x^2\right )}-\frac {99 \log \left (3+x^2\right )}{16\ 2^{2/3}}+\frac {297 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}-\frac {297 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{8\ 2^{2/3}}\\ &=-\frac {3 x^4 \left (1-x^2\right )^{2/3}}{10 \left (3+x^2\right )}+\frac {9 \left (1-x^2\right )^{2/3} \left (69+14 x^2\right )}{40 \left (3+x^2\right )}+\frac {99 \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{8\ 2^{2/3}}-\frac {99 \log \left (3+x^2\right )}{16\ 2^{2/3}}+\frac {297 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 124, normalized size = 0.93 \begin {gather*} \frac {3}{160} \left (-\frac {4 \left (1-x^2\right )^{2/3} \left (-207-42 x^2+4 x^4\right )}{3+x^2}+330 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+330 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )-165 \sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.60, size = 489, normalized size = 3.68 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 126, normalized size = 0.95 \begin {gather*} \frac {99}{32} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {3}{10} \, {\left (-x^{2} + 1\right )}^{\frac {5}{3}} - \frac {99}{64} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {99}{32} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {15}{4} \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + \frac {27 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{8 \, {\left (x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 133, normalized size = 1.00 \begin {gather*} \frac {3 \, {\left (660 \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (x^{2} + 3\right )} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 165 \cdot 4^{\frac {2}{3}} {\left (x^{2} + 3\right )} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + 330 \cdot 4^{\frac {2}{3}} {\left (x^{2} + 3\right )} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 8 \, {\left (4 \, x^{4} - 42 \, x^{2} - 207\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right )}}{320 \, {\left (x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 126, normalized size = 0.95 \begin {gather*} \frac {99}{32} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {3}{10} \, {\left (-x^{2} + 1\right )}^{\frac {5}{3}} - \frac {99}{64} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {99}{32} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {15}{4} \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + \frac {27 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{8 \, {\left (x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 148, normalized size = 1.11 \begin {gather*} \frac {99\,2^{1/3}\,\ln \left (\frac {88209\,{\left (1-x^2\right )}^{1/3}}{64}-\frac {88209\,2^{2/3}}{64}\right )}{16}+\frac {27\,{\left (1-x^2\right )}^{2/3}}{8\,\left (x^2+3\right )}+\frac {15\,{\left (1-x^2\right )}^{2/3}}{4}+\frac {3\,{\left (1-x^2\right )}^{5/3}}{10}+\frac {99\,2^{1/3}\,\ln \left (\frac {88209\,{\left (1-x^2\right )}^{1/3}}{64}-\frac {88209\,2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{32}-\frac {99\,2^{1/3}\,\ln \left (\frac {88209\,{\left (1-x^2\right )}^{1/3}}{64}-\frac {88209\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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